# Article

Ufa Mathematical Journal
Volume 13, Number 1, pp. 137-147

# On asymptotic structure of continuous-time Markov branching processes allowing immigration without higher-order moments

Imomov A.A., Meyliev A.Kh.

DOI:10.13108/2021-13-1-137

We consider a continuous-time Markov branching process allowing immigration. Our main analytical tool is the slow variation (or more general, a regular variation) conception in the sense of Karamata. The slow variation property arises in many issues, but it usually remains rather hidden. For example, denoting by $p(n)$ the perimeter of an equilateral polygon with $n$ sides inscribed in a circle with a diameter of length $d$, one can check that the function $\boldsymbol{\pi}(n):={p(n)}/d$ converges to $\pi$ in the sense of Archimedes, but it slowly varies at infinity in the sense of Karamata. In fact, it is known that $p(n)=dn\sin{\left(\pi/n\right)}$ and then it follows $\boldsymbol{\pi}(\lambda{x}) /\boldsymbol{\pi}(x) \to 1$ as $x \to \infty$ for each $\lambda > 0$. Thus, $\boldsymbol{\pi}(x)$ is so slowly approaching $\pi$ that it can be suspected that $\pi$ is not quite constant''.